New terms follow a simple rule: They’re the sum of the two preceding ones. The series starts with two 1’s, and the rest of the values proceed from there.

The Fibonacci Series has many fascinating properties, one of which is that the ratio between successive terms converges to the Golden Ratio, which is (sqrt(5)+1)/2 or about 1.618. (The ratios indeed approximate this value: 3/2 = 1.5, 5/3 = 1.666…, 8/5 = 1.6, 13/8 = 1.625, …, 55/34 ~= 1.6176 and so on.)

The Golden Ratio appears in many shapes in nature. It also occurs in a rectangle that has the following property: If a square is attached to the rectangle’s longer side to produce a new rectangle, the sides of the new rectangle have the same ratio as the old.

More specifically, suppose the old rectangle has sides of length a and b where b > a. The square is attached to the side of length b, so has dimensions b * b. The new rectangle then has sides b and a+b. The ratio between the sides of the old rectangle is b/a, and the ratio between the sides of the new is (a+b)/b = a/b + 1. If the ratios are the same, then

b/a = a/b + 1 .

Let r denote the common ratio. It follows that

r = 1/r + 1.

Equivalently, we can write

r2 – r – 1 = 0 .

The positive solution to this equation is the Golden Ratio, r = (sqrt(5)+1)/2. Its inverse is 1/r = (sqrt(5)-1)/2 = r – 1. The ratio is the only positive value with the property that value and inverse differ by 1: r is 1.618… and 1/r is 0.618.

The connection to the Fibonacci Series is that the next term is like the new side of the new rectangle: a and b leads to a+b. As the terms proceed, everything becomes golden.